Arithmetic of certain Calabi-Yau varieties and mirror symmetry.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1097.14033
author: Noriko Yui
publication date: 2001-01-01T00:00:00Z
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11F22
Mathematics Subject Classification ID: 11F23
Mathematics Subject Classification ID: 11G35
zbMATH DE Number: 1698975
zbMATH Keywords: modular forms
zbMATH Keywords: modularity conjectures
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: mirror maps
zbMATH Keywords: mirror moonshine
MaRDI profile type: Publication
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title: Arithmetic of certain Calabi-Yau varieties and mirror symmetry.
review text: This set of lecture notes serves as an overview of the arithmetic study of Calabi-Yau varieties. The article is divided into three parts; the first part discusses the modularity question for arithmetic Calabi-Yaus. The second part focuses on orbifold Calabi-Yaus and the third part focuses on \(K3\) surfaces and the mirror moonshine phenomenon. NEWLINENEWLINEFor the purposes of this article, a Calabi-Yau variety \(X\) is a smooth projective variety of some dimension \(d\) defined over a field \(K\) such that the cohomology groups \(H^i(X, \mathcal{O}_X)\) vanish for \(0 < i < d\) and the canonical sheaf \(K_X\) is trivial. Calabi-Yau varieties of dimensions 1 and 2 are elliptic curves and \(K3\) surfaces respectively. Since the focus is on the arithmetic of these varieties, we disregard the metric properties of Calabi-Yaus over \(\mathbb{C}\). NEWLINENEWLINEDue to the work of \textit{A. Wiles} [Ann. Math. (2) 141, No. 3, 443--551 (1995; Zbl 0823.11029)], \textit{R. Taylor} and \textit{A. Wiles} [Ann. Math. (2) 141, No. 3, 553--572 (1995; Zbl 0823.11030)], and \textit{C. Breuil, B. Conrad, F. Diamond} and \textit{R. Taylor} [J. Am. Math. Soc. 14, No. 4, 843--939 (2001; Zbl 0982.11033)], it is known that every elliptic curve \(E\) defined over \(\mathbb{Q}\) is modular. Among other things, this means that the \(L\)-series of the two-dimensional Galois representation \(H^1_{\text{ét}}(X, \mathbb{Q}_l)\) is the Mellin transform of a modular form of weight 2 and level \(N\) equal to the conductor of \(E\). It is conjectured that Calabi-Yau threefolds \(X\) which are rigid (\(b^3 = 2\)) are also modular; that is, the \(L\)-series of \(H^3_{\text{ét}}(X, \mathbb{Q}_l)\) (up to some factors at the primes of bad reduction of \(X\)) is the Mellin transform of a modular form of weight 4 and level \(N\) divisible only by the bad primes of \(X\). If \(X\) is not rigid, we may ask if \(H^3(X, \mathbb{Q}_l)\) splits into motivic factors \(\mathcal{I} \oplus \mathcal{M}\) where the …
description: scientific article; zbMATH DE number 1698975